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Can we just make this in a closed category? The monoidal operation doesn’t appear to be used here.
Sure. But closed monoidal categories are far more common in practice, and since any closed category embeds in a closed monoidal category by a fully faithful closed functor, the extra generality doesn’t seem significant to me.
A lot of categories are more obviously closed than monoidal. It seems strange to ask the reader to consider whether has a tensor product compatible with its closed structure when discussing the dual, when this has nothing directly to do with it.
As I say, Toby, go ahead and change it if you like.
I’ve changed it enough to be happy. (^_^)
It seems to me that it would be less circular to say that in a star-autonomous category the dual of is where is the “dualizing object” (rather than “the dual of the monoidal unit”).
Also, I find it very confusing that “the dual” of , in the sense of this new page, is not necessarily “a dual” of , in the sense of the page dual object. Is there anything we can do about that?
is the “dualizing object”
Yeah, I just blanked out on this bit of standard terminology when I wrote the circular phrase. Fixed.
Also, I find it very confusing that “the dual” of A, in the sense of this new page, is not necessarily “a dual” of A, in the sense of the page dual object.
Several pages need some fixing up. I’ll see what I can do, and report back.
Okay, made some clarifying/disambiguating remarks at dual object in a closed category (note the change of title), dualizable object, category with duals (adding, after -autonomous category, closed categories which again “do not really belong in the list”), and most significantly at predual. At the last, I edited to say that the term is most commonly encountered in the sense of “dual object in a closed category”, and that for other monoidal senses of dual it boils down to some other differently named concept.
In other words, I followed what Zoran called (rather sweetly, I thought) “Urs’s principle”, making changes and reporting back. Please feel free to improve, or say something here if you disagree with the edits.
Edit: could be a cache bug problem; dual object in a closed monoidal category isn’t redirecting to dual object in a closed category, as I hoped it would.
That helps; thanks!
could be a cache bug problem
It is; you can tell if you try to edit, which will bounces you to the home page if it’s the cache bug.
I have cleared the cache of dual object in a closed category.
It’s the other one that needed to be cleared (but it has been, or cleared itself). (Also, do we really want a table of contents with only one entry? An entry whose header doesn’t fully match its content either.)
(Also, do we really want a table of contents with only one entry? An entry whose header doesn’t fully match its content either.)
In dual object in a closed category, I changed the section title from “Definition” to “Conventional Meaning” (that’s the closest description that occurred to me, but feel free to change).
I guess Urs, by creating a table of contents, is simply preparing for a future where there is more than one section. It looks a little funny now, maybe, but I think it does no harm. :-)
It’s the other one that needed to be cleared
You mean the one of dual object in a closed monoidal category? I would have thought so, too, but it turns out that’s not how the system works: the cache file of that entry didn’t exist when I went to remove cache files. I guess the cache file is correctly being renamed when the entry is being renamed, but then something else goes wrong.
I guess Urs, by creating a table of contents, is simply preparing for a future where there is more than one section.
Yes. And now there are two sections. In fact, I would be inclined to restructure the text at “Conventional meaning” into “Definition” and “Properties - Relation to other notions” and “Examples and counterexamples”. But I won’t do that right now.
I guess the cache file is correctly being renamed when the entry is being renamed, but then something else goes wrong.
That’s funny, that’s not how it used to go wrong. But they also used to clear themselves after a while, which is what I thought must have happened here.
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